Financial Models with Levy Processes and Volatility Clustering

Transcription

1 Financial Models with Levy Processes and Volatility Clustering SVETLOZAR T. RACHEV # YOUNG SHIN ICIM MICHELE LEONARDO BIANCHI* FRANK J. FABOZZI WILEY John Wiley & Sons, Inc.

2 Contents Preface About the Authors xv xix CHAPTER 1 Introduction The Need for Better Financial Modeling of Asset Prices The Family of Stable Distribution and Its Properties Parameterization of the Stable Distribution Desirable Properties of the Stable Distributions Considerations in the Use of the Stable Distribution Option Pricing with Volatility Clustering Non-Gaussian GARCH Models Model Dependencies Monte Carlo Organization of the Book 14 References 15 CHAPTER 2 Probability Distributions Basic Concepts Discrete Probability Distributions Bernoulli Distribution Binomial Distribution " Poisson Distribution Continuous Probability Distributions Probability Distribution Function, Probability Density Function, and ' Cumulative Distribution Function Normal Distribution Exponential Distribution Gamma Distribution 28 vii

3 VIM CONTENTS Variance Gamma Distribution Inverse Gaussian Distribution Statistic Moments and Quantiles Location Dispersion Asymmetry Concentration in Tails Statistical Moments Quantiles Sample Moments " Characteristic Function Joint Probability Distributions Conditional Probability Joint Probability Distribution Defined Marginal Distribution Dependence of Random Variables Covariance and Correlation Multivariate Normal Distribution Elliptical Distributions Copula Functions Summary 54 References 54 CHAPTER 3 Stable and Tempered Stable Distributions a-stable Distribution Definition of an a-stable Random Variable Useful Properties of an a-stable Random Variable Smoothly Truncated Stable Distribution Tempered Stable Distributions Classical Tempered Stable Distribution Generalized Classical Tempered Stable Distribution Modified Tempered Stable Distribution Normal Tempered Stable Distribution Kim-Rachev Tempered Stable Distribution Rapidly Decreasing Tempered Stable Distribution Infinitely Divisible Distributions Exponential Moments Summary 82

4 Contents ix 3.5 Appendix The Hypergeometric Function The Confluent Hypergeometric Function 83 References 84 CHAPTER 4 Stochastic Processes in Continuous Time Some Preliminaries Poisson Process, Compounded Poisson Process Pure Jump Process Gamma Process Inverse Gaussian Process Variance Gamma Process a-stable Process >5 Tempered Stable Process Brownian Motion 95 A A.I Arithmetic Brownian Motion Geometric Brownian Motion Time-Changed Brownian Motion Variance Gamma Process Normal Inverse Gaussian Process Normal Tempered Stable Process Levy Process Summary 105 References 106 CHAPTER 5 Conditional Expectation and Change of Measure Events, a -Fields, and Filtration, Conditional Expectation Change of Measures Equivalent Probability Measure Change of Measure for Continuous-Time Processes Change of Measure in Tempered Stable Processes Summary 121 References 121 CHAPTERS Exponential Levy Models Exponential Levy Models 123

5 CONTENTS 6.2 Fitting a-stable and Tempered Stable Distributions Fitting the Characteristic Function Maximum Likelihood Estimation with Numerical Approximation of the Density Function Assessing the Goodness of Fit Illustration: Parameter Estimation for Tempered Stable Distributions Summary Appendix: Numerical Approximation of Probability Density and Cumulative Distribution Functions Numerical Method for the Fourier Transform 139 References 140 CHAPTER 7 Option Pricing in Exponential Levy Models Option Contract Boundary Conditions for the Price of an Option No-Arbitrage Pricing and Equivalent Martingale Measure Option Pricing under the Black-Scholes Model European Option Pricing under Exponential Tempered Stable Models Illustration: Implied Volatility Illustration: Calibrating Risk-Neutral Parameters Illustration: Calibrating Market Parameters and Risk-Neutral Parameters Together Subordinated Stock Price Model Stochastic Volatility Levy Process Model Summary 167 References 167 CHAPTER 8 Simulation Random Number Generators " Uniform Distributions Discrete Distributions Continuous Nonuniform Distributions Simulation of Particular Distributions Simulation Techniques for Levy Processes Taking Care of Small Jumps Series Representation: A General Framework Rosinsky Rejection Method a-stable Processes 192

6 Contents Xl 8.3 Tempered Stable Processes Kim-Rachev Tempered Stable Case Classical Tempered Stable Case Tempered Infinitely Divisible Processes Rapidly Decreasing Tempered Stable Case Modified Tempered Stable Case Time-Changed Brownian Motion Classical Tempered Stable Processes Variance Gamma and Skewed Variance Gamma Processes Normal Tempered Stable Processes Normal Inverse Gaussian Processes Monte Carlo Methods Variance Reduction Techniques A Nonparametric Monte Carlo Method A Monte Carlo Example 216 Appendix 217 References 220 CHAPTERS Mum-Tail f-distribution Introduction Principal Component Analysis Principal Component Tail Functions Density of a Multi-Tail t Random Variable Estimating Parameters Estimation of the Dispersion Matrix Estimation of the Parameter Set Empirical Results Comparison to Other Models, Two-Dimensional Analysis Multi-Tail t Model Check for the DAX Summary 244 References 246 CHAPTER 10 Non-Gaussian Portfolio Allocation Introduction Multifactor Linear Model Modeling Dependencies Average Value-at-Risk Optimal Portfolios 255

7 XII 10.6 The Algorithm 10.7 An Empirical Test 10.8 Summary References CHAPTER 11 Normal GARCH models 11.1 Introduction 11.2 GARCH Dynamics with Normal Innovation 11.3 Market Estimation 11.4 Risk-Neutral Estimation Out-of-Sample Performance 11.5 Summary References CONTENTS CHAPTER 12 Smoothly Truncated Stable GARCH Models 12.1 Introduction 12.2 A Generalized NGARCH Option Pricing Model 12.3 Empirical Analysis Results under the Objective Probability Measure Explaining S&P 500 Option Prices 12.4 Summary References CHAPTER 13 Infinitely Divisible GARCH Models 13.1 Stock Price Dynamic 13.2 Risk-Neutral Dynamic 13.3 Non-Normal Infinitely Divisible GARCH Classical Tempered Stable Model Generalized Tempered Stable Model Kim-Rachev Model Rapidly Decreasing Tempered Stable Model Inverse Gaussian Model Skewed Variance Gamma Model Normal Inverse Gaussian Model 13.4 Simulate Infinitely Divisible GARCH Appendix References

8 Contents XlH CHAPTER 14 Option Pricing with Monte Carlo Methods Introduction Data Set Market Estimation Performance of Option Pricing Models In-Sample Out-of-Sample Summary 355 References 356 CHAPTER 15 American Option Pricing with Monte Carlo Methods American Option Pricing in Discrete Time The Least Squares Monte Carlo Method LSM Method in GARCH Option Pricing Model Empirical Illustration Summary 372 References 372 Index 373